On a Characterization of Spartan Graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916681290350592 |
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| author | Misra, Neeldhara Nanoti, Saraswati Girish |
| author_facet | Misra, Neeldhara Nanoti, Saraswati Girish |
| contents | The eternal vertex cover game is played between an attacker and a defender on an undirected graph $G$. The defender identifies $k$ vertices to position guards on to begin with. The attacker, on their turn, attacks an edge $e$, and the defender must move a guard along $e$ to defend the attack. The defender may move other guards as well, under the constraint that every guard moves at most once and to a neighboring vertex. The smallest number of guards required to defend attacks forever is called the eternal vertex cover number of $G$, denoted $evc(G)$.
For any graph $G$, $evc(G)$ is at least the vertex cover number of $G$, denoted $mvc(G)$. A graph is Spartan if $evc(G) = mvc(G)$. It is known that a bipartite graph is Spartan if and only if every edge belongs to a perfect matching. We show that the only König graphs that are Spartan are the bipartite Spartan graphs. We also give new lower bounds for $evc(G)$, generalizing a known lower bound based on cut vertices. We finally show a new matching-based characterization of all Spartan graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06832 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Characterization of Spartan Graphs Misra, Neeldhara Nanoti, Saraswati Girish Discrete Mathematics Combinatorics The eternal vertex cover game is played between an attacker and a defender on an undirected graph $G$. The defender identifies $k$ vertices to position guards on to begin with. The attacker, on their turn, attacks an edge $e$, and the defender must move a guard along $e$ to defend the attack. The defender may move other guards as well, under the constraint that every guard moves at most once and to a neighboring vertex. The smallest number of guards required to defend attacks forever is called the eternal vertex cover number of $G$, denoted $evc(G)$. For any graph $G$, $evc(G)$ is at least the vertex cover number of $G$, denoted $mvc(G)$. A graph is Spartan if $evc(G) = mvc(G)$. It is known that a bipartite graph is Spartan if and only if every edge belongs to a perfect matching. We show that the only König graphs that are Spartan are the bipartite Spartan graphs. We also give new lower bounds for $evc(G)$, generalizing a known lower bound based on cut vertices. We finally show a new matching-based characterization of all Spartan graphs. |
| title | On a Characterization of Spartan Graphs |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2504.06832 |